A. Yu. Volkov

Strongly Coupled Quantum Discrete Liouville Theory. I: Algebraic Approach and Duality

Abstract: The quantum discrete Liouville model in the strongly coupled regime, 1 < c < 25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is self-dual: there are two exponential fields related by Hermitian conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables.

Abelian current algebra and the Virasoro algebra on the lattice

Abstract We describe how a natural lattice analogue of the abelian current algebra combined with free discrete time dynamics gives rise to the lattice Virasoro algebra and a corresponding hierarchy of conservation laws.

Hirota equation as an example of an integrable symplectic map

The Hamiltonian formalism is developed for the sine-Gordon model on the spacetime light-like lattice, first introduced by Hirota. The evolution operator is explicitly constructed in the quantum variant of the model and the integrability of the corresponding classical finite-dimensional system is established.

Yang-baxterization of the quantum dilogarithm

A new solution of the Yang-Baxter equation with spectral parameter is found. The resulting R-matrix R(x) is an operator inH⊗H, whereH=L2(ℝ). This R-matrix is required to justify the solution of the sine-Gordon model on the discrete space-time. Bibliography: 18 titles.

Discrete evolution for the zero modes of the quantum Liouville model

The dynamical system for the zero modes of the Liouville model, which is separated from the full dynamics for the discrete shifts of time t → t + π, is investigated. The structure of the modular double in the quantum case is introduced.

Hamiltonian interpretation of the Volterra model

We consider a differential-difference Volterra system in the new Hamiltonian formulation of L. A. Takhtadzhyan and L. D. Faddeev. The connection with the quantum method of the inverse problem and conformally invariant two-dimensional field theory is discussed.

A discrete version of the Landau-Lifshits equation

A discrete version of the Landau-Lifshits equation from the theory of ferromagnetism is investigated within the framework of the method of the inverse-scattering problem. Variations of action-angle type are constructed, and the energy spectrum of the model is described. The procedure of “dressing” is used to obtain the simplest soliton solution.

The new results on lattice deformation of current algebra

The topic “Quantum Integrable Models” was reviewed in the literature and presented to the conferences and schools many times. Only the reports of our own have been done on quite a few occasions (see, e.g., [1], [2]). So here we shall try to present a fresh approach to the description of the ingredients of construction of integrable models. It has gradually evolved in the process of our joint work. Whereas our goal was the Sugawara construction for the lattice affine algebra (known now as the St.Petersburg algebra), (see, e.g., [1]), some technical developments happen to be new and useful for the already developed subjects. Here we shall underline this development.

Liouville's equation and the lattice sinh-Gordon model

The equivalence of the Liouville lattice model and the massless version of the sinh-Gordon lattice model is established. Thus Liouvilles equation is included in the class of systems to which the quantum method of the inverse problem is applicable. A Hamiltonian of the quantum lattice model is constructed.